A simple closed-form formula for the conditional moments of the Ornstein-Uhlenbeck process

This paper derives a simple closed-form formula for the n th conditional moment of the Ornstein-Uhlenbeck (O-U) process, for any positive integer n. The system of recursive ordinary differential equations (ODEs) associated with the n th conditional moment of the O-U process is solved analytically. We also provide practitioners a pseudocode for an algorithm to compute the conditional moments and discuss the efficiency of our formula compared to solving the system of recursive ODEs using the direct method.